Symmetry and symmetry breaking for the fractional Caffarelli-Kohn-Nirenberg inequality

TL;DR

This paper investigates the fractional Caffarelli-Kohn-Nirenberg inequality, focusing on the existence, nonexistence, symmetry, and symmetry breaking of extremal solutions, revealing novel behaviors and establishing new methods for analyzing non-local equations.

Contribution

It introduces new results on symmetry breaking, extremal solutions, and a novel proof technique for non-degeneracy in fractional inequalities, extending classical methods to non-local operators.

Findings

Existence and nonexistence regions for extremal solutions

Identification of a Felli-Schneider type curve for symmetry breaking

New proof of non-degeneracy applicable to general operators

Abstract

In this paper, we will consider the fractional Caffarelli-Kohn-Nirenberg inequality \begin{equation*} {\Lambda} \left(\int_{\mathbb R^n}\frac{|u(x)|^{p}}{|x|^{{\beta} {p}}}\,dx\right)^{\frac{2}{p}}\leq \int_{\mathbb R^n}\int_{\mathbb R^n}\frac{(u(x)-u(y))^2}{|x-y|^{n+2\gamma}|x|^{{\alpha}}|y|^{{\alpha}}}\,dy\,dx \end{equation*} where $\gamma\in(0,1)$, $n\geq 2$, and $\alpha,\beta\in\mathbb R$ satisfy \begin{equation*} \alpha\leq \beta\leq \alpha+\gamma, \ -2\gamma<\alpha<\frac{n-2\gamma}{2}, \end{equation*} and the exponent $p$ is chosen to be \begin{equation*} p=\frac{2n}{n-2\gamma+2(\beta-\alpha)}, \end{equation*} such that the inequality is invariant under scaling. We first study the existence and nonexistence of extremal solutions. Our next goal is to show some results on the symmetry and symmetry breaking region for the minimizers; these suggest the existence of a Felli-Schneider type curve separating both regions but, surprisingly, we find a novel behavior as $\alpha\to -2\gamma$. The main idea in the proofs, as in the classical case, is to reformulate the fractional Caffarelli-Kohn-Nirenberg inequality in cylindrical variables. Then, in order to find the radially symmetric solutions we need to solve a non-local ODE. For this equation we also get uniqueness of minimizers in the radial symmetry class; indeed, we show that the unique continuation argument of Frank-Lenzmann (Acta'13) can be applied to more general operators with good spectral properties. We provide, in addition, a completely new proof of non-degeneracy which works for all critical points. It is based on the variation of constants approach and the non-local Wronskian of Ao-Chan-DelaTorre-Fontelos-Gonz\'alez-Wei (Duke'19).